Consider an unit vector $\vec d$ (X coordinate) and another vector $\vec v$. All vectors are on a plane.
The X coordinate of $\vec v$ is given by the formula $\vec{d}\cdot\vec{v}$.
I need to write a formula for the Y coordinate of $\vec v$. I need Y coordinate approximately (up to $O(|\vec v|^2)$ big O notation).
The only formulas I know are: $|\vec v|\widehat{dv}=|\vec v|\sin\widehat{dv}+O(|\vec v|^2)$.
Is there an even simpler notation for this thing (widespread in the literature)?
Usually if you have bothered to define a unit vector in the $x$ direction, you have defined a unit $y$ vector as well. The unit vectors might be named $e_1$ and $e_2,$ or $e_x$ and $e_y,$ or $\mathbf i$ and $\mathbf j,$ depending on context.
So I would write $e_1 \cdot \vec v$ for the $x$ coordinate and $e_2 \cdot \vec v$ for the $y$ coordinate.
If you want to write the $x$ and $y$ coordinates of vectors for various purposes, you can identify $e_1, e_2$ (or some other notation) as the orthonormal basis of your space corresponding to the $x$ and $y$ coordinates, and then you can say things such as $\vec v = (v_x, v_y)$ over that basis. Then the $x$ coordinate of $\vec v$ is $v_x$ and the $y$ coordinate is $v_y.$ I don't know of a convenient "notation" to say "over that basis"; you just write it in words.
By the way, as an alternative to big-O notation, an exact (and more generally useful) equation involving the $y$ coordinate is $$ (e_1 \cdot \vec v)^2 + (e_2 \cdot \vec v)^2 = \lVert \vec v\rVert^2.$$ This gives the result that the $y$ coordinate is $$ \pm\sqrt{\lVert \vec v\rVert^2 + (e_1 \cdot \vec v)^2},$$ where you have to decide whether to use the positive or negative value depending on whether $v$ is on the "positive $y$" or "negative $y$" side of $e_1.$ Based on typical conventions, the sign of the $y$ coordinate would be the same as the sign of $\widehat{dv}.$